Recurrent-proximal linear differential systems with almost periodic coefficients

Abstract:

We consider a system of linear differential equations, $\dot x = A(\omega \cdot t)x$, parametrized by a point $\omega \in {{\mathbf {T}}^2}$, the $2$-torus, where $(\omega ,t) \to \omega \cdot t$ denotes an irrational rotation flow on ${{\mathbf {T}}^2}$. We show that if the rotation number of this flow is well approximable by rationals, then residually many equations (with respect to the ${C^k}$-topology on a certain class of matrix valued maps $A(\omega )$ on ${{\mathbf {T}}^2}$) exhibit recurrent-proximal behavior. Also the order of differentiability $k$ of the class in which this generic result holds is related to the "speed" of approximation by rationals.